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MATH32052 HYPERBOLIC GEOMETRY
创建日期：2024-08-27 14:52:34
所属分类：数学mathematics
标签： MATH32052
MATH32052 HYPERBOLIC GEOMETRY

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MATH32052


HYPERBOLIC GEOMETRY

Notation: Throughout, H denotes the upper half-plane, ?H denotes the boundary of H, D denotes the Poincar′e disc, and ?D denotes the boundary of D. We also write M¨ob(H) for the group of all M¨obius transformations of H and M¨ob(D) for the group of all M¨obius transformations of D.

You may use the fact that if 0 < a < b then dH(ia, ib) = log b/a. (Here and throughout this exam, log denotes the natural logarithm.)

You may also use the fact that for any z, w ∈ H we have the formula

1.

(i) The equation of a straight line in R 2 is given by ax + by + c = 0, where a, b, c ∈ R are constants. Let z = x + iy. Show that the equation of this straight line can be written in C in the form.

for suitable β ∈ C, γ ∈ R.

Determine β, γ in terms of a, b, c.

[6 marks]

(ii) Let m ∈ R, m ≠ 0. Consider the straight line in R 2 given by y = mx. Determine β and γ in terms of m for this straight line when written in C.

[2 marks]

[End of Question 1; 8 marks total]

2.

(i) Let

be a M¨obius transformation of H.

What does it mean to say that γ is normalised?

How is τ (γ) defined?

[4 marks]

(ii) Consider the following two statements:

(a) τ (γ?1) = τ (γ) for all γ ∈ M¨ob(H).

(b) τ (γ1γ2) = τ (γ1)τ (γ2) for all γ1, γ2 ∈ M¨ob(H).

In each case, decide whether the statement is true or false. If the statement is true then give a proof; if the statement is false then give a counterexample.

[8 marks]

[End of Question 2; 12 marks total]

3. Throughout this question you may use the facts that:

(i) Consider the hyperbolic triangles illustrated in Figure 1. In (a), an arbitrary hyperbolic triangle with one ideal vertex and one right-angle is drawn. In (b) a hyperbolic triangle with one ideal vertex at ∞ and one right-angle at i is drawn. The other internal angle in both (a) and (b) is θ and the side with finite hyperbolic length has length a.

The Angle of Parallelism formula states that for a triangle such as in Figure 1(a) we have

Explain briefly why, when proving (2), we can assume without loss of generality that the triangle is as illustrated in Figure 1(b).

Explain why, for the point s + it in Figure 1(b), we have s 2 + t 2 = 1.

Use Figure 1(b) to prove the Angle of Parallelism formula (2).

Figure 1: (a) An arbitrary hyperbolic triangle with one ideal vertex and internal angles π/2 and θ. (b) A hyperbolic triangle with one ideal vertex at ∞ and right-angle at i.

[10 marks]

(ii) Consider the hyperbolic quadrilateral illustrated in Figure 2. In this quadrilateral, two sides are vertical.

Figure 2: A hyperbolic quadrilateral with two vertical sides.

Use Euclidean geometry to show that the angle ? is equal to 3/2π.

Consider the vertex at u + iv. Determine the values of u and v.

[8 marks]

(iii) Use the results of (i) and (ii) to show that the perimeter of the quadrilateral in Figure 2 has hyperbolic length

[8 marks]

[End of Question 3; 26 marks total]

4.

(i) Let z1, z2 ∈ H, z1 ≠ z2. Let [z1, z2] denote the arc of geodesic between z1 and z2. Recall that the perpendicular bisector of [z1, z2] is the unique geodesic in H that passes through the hyperbolic midpoint of [z1, z2] at right angles.

Prove that the perpendicular bisector of [z1, z2] can also be described by

{z ∈ H | dH(z, z1) = dH(z, z2)}                                                        (3)

[10 marks]

(ii) Let n ∈ Z, n ≠ 0. Let z1 = 3i and z2 = 4n + 3i. Using the formula in (3), determine the perpendicular bisector of [z1, z2].

[6 marks]

Let Γ ? M¨ob(H) be a Fuchsian group. Let p ∈ H be such that γ(p) ≠ p for all γ ∈ Γ \ {id}. Recall that the Dirichlet polygon D(p) is defined as follows:

? Let γ ∈ Γ \ {id}. Let Lp(γ) denote the perpendicular bisector of [p, γ(p)].

? Let Hp(γ) denote the half-plane determined by Lp(γ) that contains p.

? Let D(p) = ∩γ∈Γ\{id} Hp(γ).

(iii) Let Γ = {γn | γn(z) = z + 4n, n ∈ Z}. Let p = 3i. Using the result you obtained in (ii), determine the Dirichlet polygon D(p).

[6 marks]

(iv) The set D(p) is a fundamental domain for Γ. Sketch the corresponding tiling of H.

[2 marks]

[End of Question 4; 24 marks total]

5.

(i) Let E be an elliptic cycle. What does it mean to say that E satisfies the elliptic cycle condition?

Let P be a parabolic cycle. What does it mean to say that P satisfies the parabolic cycle condition?

[4 marks]

(ii) Consider the hyperbolic quadrilateral as illustrated in Figure 3. Let λ ∈ R, λ > 0, and define

Figure 3: A hyperbolic quadrilateral. The internal angle at i √ 3/2 is 2π/3.

Show that, for all λ > 0, the transformation γ1 maps s1 to s2. (You may use any results from the course, provided that you state them clearly.)

[4 marks]

(iii) Determine the parabolic cycle that contains the vertex ∞.

Show that there is exactly one value of λ that ensures that the parabolic cycle condition holds for this parabolic cycle and determine this value of λ.

[6 marks]

(iv) Determine all remaining elliptic and parabolic cycles. For each elliptic cycle, determine the elliptic cycle transformation and the angle sum along the elliptic cycle. For each parabolic cycle, determine the parabolic cycle transformation.

[10 marks]

(v) For the value of λ you determined in (iii), use Poincar′e’s Theorem to show that γ1, γ2 gener-ate a Fuchsian group Γ. Give a presentation of this group in terms of generators and relations.

[2 marks]

(vi) For the Fuchsian group determined in (iv), briefly describe the quotient space H/Γ.

[4 marks]

[End of Question 5; 30 marks total]

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